What it shows
Eratosthenes found the size of the Earth by comparing noon shadows at two places a known distance apart. You can repeat it with a friendly school in another city to find the Earth’s circumference.
The science
The Sun is so far away that its rays reach the Earth almost parallel. At the same solar noon on the same day, the Sun’s angle from the vertical is different at two places on the same north-south line, because the ground curves. The difference in angle between the two shadows equals the angle between the two places at the centre of the Earth. If this angle is d degrees for a distance S on the ground, the whole circumference is S x 360 / d. A stick of height h and shadow s gives the angle from the vertical by tan angle = s / h.
Materials
Cost: free. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- A straight vertical stick, 1 m, set up with a plumb line
- A level piece of ground or a board
- A tape measure and a pencil
- A partner school in a city due north or south, at least 300 km away
- A map or distance table between the two cities
- A watch showing correct local time
- Graph paper
Safety
For this project: Do not look at the Sun. Measure only the shadow. Work with an adult in an open, safe area.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
Step by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Find the date of the test and choose one that both schools can use, such as a clear day around an equinox in March or September.
Find the local solar noon, the time when the shadow of your stick is shortest. Start measuring ten minutes earlier and mark the tip of the shadow every five minutes.
At the shortest shadow, measure the shadow length s and the height h of the stick. Repeat three times and take the average.
Calculate the angle of the Sun from the vertical with tan inverse of (s / h).
Your partner school does the same on the same day and sends the angle.
Find the difference between the two angles, which is the angle between the two places at the Earth’s centre. Use the larger minus the smaller if both are on the same side of the Sun.
Measure the north-south distance between the two places on a map and calculate the Earth’s circumference as distance x 360 / angle difference.
Compare your value with the accepted value of about 40,000 km and write the percentage error and its likely causes.
What you should see
An estimate of the Earth’s circumference that is within about 10 to 20 per cent of 40,000 km, depending on the accuracy of the shadow, the distance and how north-south the two places are.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
01Why do shadows at two places differ at the same time?
The Earth is curved, so the Sun’s rays strike the ground at different angles.
02Why must the two places be on the same north-south line?
Otherwise the difference in angle also includes a difference in the time of noon.
03Why are the Sun’s rays nearly parallel?
The Sun is about 150 million km away, a distance huge compared with the Earth’s width of about 12,700 km.
04What did Eratosthenes use for the distance?
The distance between Syene and Alexandria, which he took from travellers’ estimates.
Take it further
Use several partner schools at different distances and plot the angle difference against the distance to show a straight line.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
Free ebooks for this project
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